Showing (a) is straightforward: simply take \(D = \emptyset \) . \(\Pi _1\)  has a CWV
(via correctness of the reduction in [1]} as explained above),
whereas \(\Pi _2\)  does not have a CWV, since the atom \({\mathit {sat}}\)  is always
derived in any GL-reduct of \(\Pi _2\) , destroying each potential CWV. Showing
(b) is a little more involved. We will show this by contradiction. To this
end, assume that the 3-QBF is unsatisfiable, but some set of facts \(D\subseteq \mathcal {A}\)  exists, such that \(\Pi _1 \cup D\)  and \(\Pi _2 \cup D\)  are not
equivalent. Our plan is to show that \(D\)  cannot contain any atoms from
\(\mathcal {A}\) , but also cannot be empty. Let us look at the atoms in \(\mathcal {A}\)  in
turn.
